The Hadamard product and the free convolutions
Article Type
Research Article
Publication Title
Statistics and Probability Letters
Abstract
It is shown that if a probability measure ν is supported on a closed subset of (0,∞), that is, its support is bounded away from zero, then the free multiplicative convolution of ν and the semicircle law is absolutely continuous with respect to the Lebesgue measure. For the proof, a result concerning the Hadamard product of a deterministic matrix and a scaled Wigner matrix is proved and subsequently used. As a byproduct, a result, showing that the limiting spectral distribution of the Hadamard product is same as that of a symmetric random matrix with entries from a mean zero stationary Gaussian process, is obtained.
First Page
150
Last Page
157
DOI
10.1016/j.spl.2017.04.009
Publication Date
8-1-2017
Recommended Citation
Chakrabarty, Arijit, "The Hadamard product and the free convolutions" (2017). Journal Articles. 2475.
https://digitalcommons.isical.ac.in/journal-articles/2475
Comments
Open Access, Green